Search results for "Applied Mathematics"
showing 10 items of 4379 documents
Computation of travelling wave solutions of scalar conservation laws with a stiff source term
2003
Abstract In this paper we propose a nonoscillatory numerical technique to compute the travelling wave solution of scalar conservation laws with a stiff source term. This procedure is based on the dynamical behavior described by the associated stationary ODE and it reduces/avoids numerical errors usually encountered with these problems, i.e., spurious oscillations and incorrect wave propagation speed. We combine this treatment with either the first order Lax–Friedrichs scheme or the second order Nessyahu–Tadmor scheme. We have tested several model problems by LeVeque and Yee for which the stiffness coefficient can be increased. We have also tested a problem with a nonlinear flux and a discon…
Approximate Lax–Wendroff discontinuous Galerkin methods for hyperbolic conservation laws
2017
Abstract The Lax–Wendroff time discretization is an alternative method to the popular total variation diminishing Runge–Kutta time discretization of discontinuous Galerkin schemes for the numerical solution of hyperbolic conservation laws. The resulting fully discrete schemes are known as LWDG and RKDG methods, respectively. Although LWDG methods are in general more compact and efficient than RKDG methods of comparable order of accuracy, the formulation of LWDG methods involves the successive computation of exact flux derivatives. This procedure allows one to construct schemes of arbitrary formal order of accuracy in space and time. A new approximation procedure avoids the computation of ex…
Riemann solvers in relativistic astrophysics
1999
AbstractOur contribution reviews High Resolution Shock Capturing methods (HRSC) in the field of relativistic hydrodynamics with special emphasis on Riemann solvers. HRSC techniques achieve highly accurate numerical approximations (formally second order or better) in smooth regions of the flow, and capture the motion of unresolved steep gradients without creating spurious oscillations. One objective of our contribution is to show how these techniques have been extended to relativistic hydrodynamics, making it possible to explore some challenging astrophysical scenarios. We will review recent literature concerning the main properties of different special relativistic Riemann solvers, and disc…
Adaptive mesh refinement techniques for high-order shock capturing schemes for multi-dimensional hydrodynamic simulations
2006
The numerical simulation of physical phenomena represented by non-linear hyperbolic systems of conservation laws presents specific difficulties mainly due to the presence of discontinuities in the solution. State of the art methods for the solution of such equations involve high resolution shock capturing schemes, which are able to produce sharp profiles at the discontinuities and high accuracy in smooth regions, together with some kind of grid adaption, which reduces the computational cost by using finer grids near the discontinuities and coarser grids in smooth regions. The combination of both techniques presents intrinsic numerical and programming difficulties. In this work we present a …
Flux-gradient and source-term balancing for certain high resolution shock-capturing schemes
2009
Abstract We present an extension of Marquina’s flux formula, as introduced in Fedkiw et al. [Fedkiw RP, Merriman B, Donat R, Osher S. The penultimate scheme for systems of conservation laws: finite difference ENO with Marquina’s flux splitting. In: Hafez M, editor. Progress in numerical solutions of partial differential equations, Arcachon, France; July 1998], for the shallow water system. We show that the use of two different Jacobians at cell interfaces prevents the scheme from satisfying the exact C -property [Bermudez A, Vazquez ME. Upwind methods for hyperbolic conservation laws with source terms. Comput Fluids 1994;23(8):1049–71] while the approximate C -property is satisfied for high…
Large time behavior for a porous medium equation in a nonhomogeneous medium with critical density
2014
Abstract We study the large time behavior of solutions to the Cauchy problem for the porous medium equation in nonhomogeneous media with critical singular density | x | − 2 ∂ t u = Δ u m , in R N × ( 0 , ∞ ) , where m > 1 and N ≥ 3 , with nonnegative initial condition u ( x , 0 ) = u 0 ( x ) ≥ 0 . The asymptotic behavior proves to have some interesting and striking properties. We show that there are different asymptotic profiles for the solutions, depending on whether the continuous initial data u 0 vanishes at x = 0 or not. Moreover, when u 0 ( 0 ) = 0 , we show the convergence towards a peak-type profile presenting a jump discontinuity, coming from an interesting asymptotic simplification…
A system of conservation laws with discontinuous flux modelling flotation with sedimentation
2019
Abstract The continuous unit operation of flotation is extensively used in mineral processing, wastewater treatment and other applications for selectively separating hydrophobic particles (or droplets) from hydrophilic ones, where both are suspended in a viscous fluid. Within a flotation column, the hydrophobic particles are attached to gas bubbles that are injected and float as aggregates forming a foam or froth at the top that is skimmed. The hydrophilic particles sediment and are discharged at the bottom. The hydrodynamics of a flotation column is described in simplified form by studying three phases, namely the fluid, the aggregates and solid particles, in one space dimension. The relat…
Exact Mechanical Hierarchy of Non-Linear Fractional-Order Hereditariness
2020
Non-local time evolution of material stress/strain is often referred to as material hereditariness. In this paper, the widely used non-linear approach to single integral time non-local mechanics named quasi-linear approach is proposed in the context of fractional differential calculus. The non-linear model of the springpot is defined in terms of a single integral with separable kernel endowed with a non-linear transform of the state variable that allows for the use of Boltzmann superposition. The model represents a self-similar hierarchy that allows for a time-invariance as the result of the application of the conservation laws at any resolution scale. It is shown that the non-linear spring…
A new mathematical tool for an exact treatment of open quantum system dynamics
2005
A new method to obtain an operatorial exact solution of a wide class of Markovian master equations is presented. Its key point is the existence of a constant of motion partitioning the Hilbert space into finite-dimensional subspaces. The consequent possibility of representing the reduced density operator as a block diagonal matrix is shown. Each “block operator” evolves under the action of a non-unitary operator explicitly derived. Our mathematical approach is illustrated applying it to simple physical systems.
Dynamic analysis for axially moving viscoelastic panels
2012
In this study, stability and dynamic behaviour of axially moving viscoelastic panels are investigated with the help of the classical modal analysis. We use the flat panel theory combined with the Kelvin–Voigt viscoelastic constitutive model, and we include the material derivative in the viscoelastic relations. Complex eigenvalues for the moving viscoelastic panel are studied with respect to the panel velocity, and the corresponding eigenfunctions are found using central finite differences. The governing equation for the transverse displacement of the panel is of fifth order in space, and thus five boundary conditions are set for the problem. The fifth condition is derived and set at the in-…